Angle Relationships

A handful of angle patterns explain most angle puzzles. When two lines cross, when a ray leaves a line, or when several rays meet at one point, the angles that appear are linked: some are equal, and some add up to 90°, 180° or 360°. If you can name the pattern, you can write an equation and solve for an unknown angle. This lesson covers adjacent, complementary, supplementary, linear-pair and vertical angles, angles around a point and angle bisectors, and then shows how to set up, solve and check an angle equation. Angles made by parallel lines and a transversal have their own lesson.

Naming and measuring angles

An angle is formed by two rays that share an endpoint. The shared endpoint is the vertex. Angles are measured in degrees (°).

To name an angle with three letters, put the vertex in the middle: ∠ABC is the angle with vertex B and sides along rays BA and BC. You can write ∠B when only one angle has that vertex, but where several angles meet, use three letters.

The symbol ∠ABC names the angle itself, the figure. The symbol m∠ABC means the measure of that angle, a number of degrees. So you write m∠ABC = 40° and read it “the measure of angle ABC is 40 degrees.” If two angles have the same measure, write m∠ABC = m∠DEF.

Angles are sorted by size. An acute angle is between 0° and 90°. A right angle is exactly 90° and is marked with a small square. An obtuse angle is between 90° and 180°. A straight angle is exactly 180°; it looks like a straight line with the vertex somewhere on it.

Adjacent angles

Two angles are adjacent if they share a vertex and a side and do not overlap. Think of two rooms that share a wall: they are neighbors.

If a ray BD lies inside ∠ABC, then ∠ABD and ∠DBC are adjacent, and their measures add up to the whole angle: m∠ABD + m∠DBC = m∠ABC.

That sum is all that adjacency promises. Adjacent angles can have any sizes. They do not have to be equal, and they do not have to add to 180°. They add to 180° only when their outer sides form a straight line, which is a linear pair (coming up).

Complementary and supplementary angles

Two angles are complementary if their measures add up to 90°. Two angles are supplementary if their measures add up to 180°. Each angle is called the complement (or the supplement) of the other.

The complement of an angle that measures x° is (90 − x)°, which needs 0 < x < 90. The supplement is (180 − x)°, which needs 0 < x < 180. For example, the complement of 38° is 52° and its supplement is 142°.

Complementary and supplementary angles do not have to be adjacent. Angles of 25° and 65° are complementary even if they sit in two different corners of a drawing. The words describe exactly two angles: three angles that happen to add to 90° are not called complementary.

A way to keep them straight: C is for Corner (90°, a square corner) and S is for Straight (180°, a straight line). Also, C comes before S in the alphabet, and 90 comes before 180.

Linear pairs

Two adjacent angles form a linear pair if their outer sides are opposite rays, so that together the two angles make a straight line. Together they make a straight angle, so a linear pair always adds to 180°: it is a supplementary pair.

Mind the direction of that statement. Every linear pair is supplementary, but not every supplementary pair is a linear pair. A 140° angle and a 40° angle drawn far apart are supplementary, but they are not adjacent, so they are not a linear pair. (If two adjacent angles do add to 180°, their outer sides form a straight line, so they are a linear pair.)

Vertical angles

When two straight lines cross, they form four angles. Vertical angles are the two angles directly opposite each other at the crossing. They share only the vertex. Two crossing lines make two pairs of vertical angles. “Vertical” does not mean up and down here; it names angles that are opposite each other at one vertex.

Vertical angles are equal. Here is why. Number the four angles around the crossing 1, 2, 3 and 4 in order. Angles 1 and 2 are a linear pair, so m∠1 = 180° − m∠2. Angles 2 and 3 are also a linear pair, so m∠3 = 180° − m∠2. Both m∠1 and m∠3 equal 180° − m∠2, so m∠1 = m∠3. The same argument shows m∠2 = m∠4.

This is a fact about two straight lines crossing. Two angles that merely look opposite in another figure, such as opposite corners of a lopsided four-sided shape, do not have to be equal.

Each angle is a linear pair with both of its neighbors. So if one angle measures 55°, the angle across from it is 55° and each neighbor is 125°.

Angles around a point

Angles that share one vertex and fill the whole space around it, with no gaps and no overlaps, add up to 360°. That is one full turn. A straight angle is half a turn (180°) and a right angle is a quarter turn (90°).

Two crossing lines make a good check. The four angles measure x°, (180 − x)°, x° and (180 − x)°, and x + (180 − x) + x + (180 − x) = 360.

Angle bisectors

An angle bisector is a ray that starts at the vertex and splits the angle into two angles with equal measures. If ray BD bisects ∠ABC, then m∠ABD = m∠DBC, and each of them is half of the whole:

m∠ABD = m∠ABC/2, so m∠ABC = 2 × m∠ABD.

The bisector lies inside the angle. Bisecting a right angle gives two 45° angles, and bisecting a straight angle gives two right angles.

Diagrams are not drawn to scale

A diagram is a sketch to help you think. Unless it says otherwise, it is not drawn to scale. Two angles that look equal may not be, and an angle that looks like 90° may not be.

What you can trust: the given measures and labels, matching equal-angle marks (the same number of small arcs or ticks), a small square marking a right angle, and, in standard geometry diagrams, that a line drawn straight is straight and that a point drawn inside an angle is inside it.

What you cannot trust: how big an angle looks, whether two angles look equal, or whether two lines look perpendicular. If a problem does not say that two angles are equal, that a ray is a bisector, or that an angle is right, do not assume it. Example 9 shows a case where the picture and the numbers disagree.

Setting up and solving angle equations

Most angle problems give the angles as expressions with an unknown x. In the equations, x stands for a number of degrees, so the degree sign is left out until you write an angle measure. Use four steps:

1. Name the pattern: complementary (sum 90), supplementary or linear pair (sum 180), around a point (sum 360), vertical angles (equal), or bisector (equal halves, and the whole is twice a half).
2. Equation: write it. Add the angles and set the sum equal to the total, or set equal angles equal to each other.
3. Solve for x.
4. Check by substituting. Find every angle, then make sure the angles fit the pattern (they add to the right total, or they are equal) and that each angle is a possible size, which means it is positive.

The value of x is not always the answer. If the question asks for an angle, substitute x back into the expression. And solving the equation is not enough on its own: if the substitution gives a negative angle, no such figure exists (Example 10).

Where this stops

This lesson uses only crossing lines, rays from one point, and sums. When a line crosses two parallel lines, more pairs of equal and supplementary angles appear. That is the next lesson, on parallel lines and transversals. The sum of the angles inside a triangle is also a separate lesson.